Question

Find the variance of numbers obtained on thrown an unbiased die.

Answer 1

$x_i$	$p_i=P\left(x_i\right)$	$p_i{x}_i$	$p_i{x_i}^2$
$1$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$
$2$	$\frac{1}{6}$	$\frac{1}{3}$	$\frac{2}{3}$
$3$	$\frac{1}{6}$	$\frac{1}{2}$	$\frac{3}{2}$
$4$	$\frac{1}{6}$	$\frac{2}{3}$	$\frac{8}{3}$
$5$	$\frac{1}{6}$	$\frac{5}{6}$	$\frac{25}{6}$
$6$	$\frac{1}{6}$	$1$	$6$
		$\sum p_i{x}_i=\frac{7}{2}$	$\sum p_i{x_i}^2=\frac{91}{6}$

Variance, $\sigma =\sum p_i{x_i}^2-{(\sum p_i{x}_i)}^2$

$\Rightarrow \sigma =\frac{91}{6}-{\left(\frac{7}{2}\right)}^2=\frac{91}{6}-\frac{49}{4}$

$\Rightarrow \sigma =\frac{35}{12}$